Due to its well-developed mathematical foundation, its connection to other ML kernel methods such as support vector machines [1]}, its applicability to domains more general than Euclidean space, and its relative ease of estimation, the distance \(\gamma _k\)  has become a powerful tool in statistical two-sample (homogeneity) testing for ML applications. Estimation of (REF ) from sample data can be achieved using \(O(n^2)\)  operations, and the sample estimate converges to the population value as \(O(n^{-1/2})\) , independently of the dimension \(d\) . Furthermore, approximations also exist that can be computed with complexity \(O(n)\) , making \(\gamma _k\)  a practical quantity for beam dynamics applications.
